The prolongation operator \(\mathbf {P}_i \in \mathbb {R}^{N \times M_i}\)  consists of the unit vectors corresponding to the \(M_i\)  indices in the \(i\) -th index block. Pre- and post-multiplying the system matrix \(\mathbf {A}\in \mathbb {R}^{N \times N}\)  with the (transpose of) this prolongation operator restricts it to the submatrix \(\mathbf {A}_i\in \mathbb {R}^{M_i \times M_i}\)  consisting of only the indices in the \(i\) -th index block. Similarly, the opposite pre- and post-multiplication with these operators injects the local inverse \(\mathbf {A}_i^{-1}\in \mathbb {R}^{M_i \times M_i}\)  into the matrix \(\mathbf {B}_i\in \mathbb {R}^{N \times N}\) . It is to be noted that the index blocks may overlap. In additive Schwarz these contributions are then added in \(\mathbf {B}_{\textsc {AS}}\)  and treated simultaneously. In multiplicative Schwarz, repetitions of the same index are treated sequentially. Application of the multiplicative Schwarz preconditioner can be expressed by recursive relations; see, e.g., [1]}. It is noted that index blocks consisting of a single index simply reduce to Jacobi preconditioning in additive Schwarz, and to Gauss-Seidel preconditioning in multiplicative Schwarz. The formal definition and details about the construction of Schwarz preconditioners are presented [2]} and [1]}.
